Finite Group Databasebeta

← #144566

Subgroups — #144566

Order 512.

287 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33298C211yesmaximal, characteristic
2#11859C212yesmaximal
3#9850C211yesmaximal, characteristic
4#7221C212yesmaximal
5#7149C211yesmaximal, characteristic
6#3170C2211yescharacteristic
7#2862—21no—
8#2163C2212yes—
9#1994—21no—
10#1993—24no—
11#1835—21no—
12#1835C2212yes—
13C4×#97C2211yescharacteristic
14#1566C2211yescharacteristic
15#1493—21no—
16#1348—24no—
17#533—21no—
18C2×C4⋊Q8—22no—
19#500Q811yescharacteristic, complemented
20#494Q811yescharacteristic, complemented
21#487—21no—
22#467—22no—
23C4×Q16—22no—
24#423—21no—
25C2×#97—22no—
26C2×C4×C8D411yescharacteristic
27C2×C4×C8C2311yescharacteristic
28#391—42no—
29#391—42no—
30#382—22no—
31#382—22no—
32C43D411yescharacteristic
33C4⋊C16—22no—
34C22⋊C16—22no—
35#346—22no—
36#340—24no—
37#340—22no—
38#340D414yes—
39#339—42no—
40#334—22no—
41Q8⋊C8—42no—
42C22×Q8—42no—
43C2×Q16—22no—
44C2×M16—41no—
45C22×C8—21no—
46C22×C8C2×D412yes—
47C4⋊Q8—21no—
48C4⋊Q8—21no—
49C4⋊Q8—48no—
50#125—41no—
51C22⋊Q8—41no—
52C22⋊Q8—41no—
53C4×Q8—41no—
54C4×Q8—41no—
55#117—41no—
56C2×C4⋊C4—24no—
57C2×C4⋊C4—42no—
58C2×C42—21no—
59C2×C42—21no—
60C2×C42D812yes—
61C2×C42C2×D411yescharacteristic
62C2×C16—42no—
63#107—22no—
64C4⋊C8—41no—
65C4⋊C8—41no—
66#103—42no—
67#97—44no—
68C4×C8—21no—
69C4×C8SD1612yes—
70C4×C8C2×Q811yescharacteristic
71C4×C8#4111yescharacteristic
72C4×C8#4112yes—
73#95—44no—
74#95—42no—
75#95—42no—
76C2×Q8—41no—
77C2×Q8—41no—
78C2×Q8—84no—
79C2×Q8—84no—
80C2×Q8—84no—
81C22×C4—22no—
82C22×C4—44no—
83C22×C4#12011yescharacteristic
84C22×C4C2×D811yescharacteristic
85C22×C4AGL(1,ℤ/8)11yescharacteristic
86Q16—82no—
87M16—82no—
88C2×C8—22no—
89C2×C8—42no—
90C2×C8—42no—
91C2×C8—81no—
92C2×C8C22⋊Q812yes—
93C2×C8#12412yes—
94C4⋊C4—48no—
95C4⋊C4—41no—
96C4⋊C4—41no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C4⋊C4—84no—
100C4⋊C4—81no—
101C4⋊C4—81no—
102C22⋊C4—81no—
103C42—24no—
104C42—42no—
105C42—81no—
106C42—81no—
107C42C22⋊Q811yescharacteristic
108C42#12411yescharacteristic
109C42C2×SD1611yescharacteristic
110C42#13711yescharacteristic
111C16—82no—
112C23#44711yescharacteristic
113Q8—82no—
114Q8—82no—
115Q8—168no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—24no—
119C2×C4—41no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4—84no—
123C2×C4—84no—
124C2×C4—84no—
125C2×C4—81no—
126C2×C4—81no—
127C2×C4#39311yescharacteristic
128C2×C4#44811yescharacteristic
129C2×C4#47411yescharacteristic
130C2×C4#47611yescharacteristic
131C2×C4#48611yescharacteristic
132C2×C4#48911yescharacteristic
133C8—22no—
134C8—22no—
135C8—42no—
136C8—161no—
137C22—21no—
138C22—21no—
139C22#202311yescharacteristic
140C22#219112yes—
141C4—21no—
142C4—21no—
143C4—21no—
144C4—21no—
145C4—21no—
146C4—44no—
147C4—164no—
148C4—161no—
149C4—161no—
150C4#203112yes—
151C2—22no—
152C2#1036211yescharacteristic
153C2#1336811yescharacteristic
154C2#1337311yescharacteristic
subgroupG/H
center, socleC22#2023
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#447
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
19. #500Q8semidirect
20. #494Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#332982
#118592
#98502
#72212
#71492